Noncommutative algebras related with Schubert calculus on Coxeter groups

For any finite Coxeter system (W,S) we construct a certain noncommutative algebra, the so-called bracket algebra, together with a family of commuting elements, the so-called Dunkl elements. The Dunkl elements conjecturally generate an algebra which is canonically isomorphic to the coinvariant algebra of the Coxeter group W. We prove this conjecture for classical Coxeter groups and I₂(m). We define a “quantization” and a multiparameter deformation of our construction and show that for Lie groups of classical type and G₂, the algebra generated by Dunkl’s elements in the quantized bracket algebra is canonically isomorphic to the small quantum cohomology ring of the corresponding flag variety, as described by B. Kim. For crystallographic Coxeter systems we define the so-called quantum Bruhat representation of the corresponding bracket algebra. We study in more detail the structure of the relations in B_n-, D_n- and G₂-bracket algebras, and as an application, discover a Pieri-type formula in the B_n-bracket algebra. As a corollary, we obtain a Pieri-type formula for multiplication of an arbitrary B_n-Schubert class by some special ones. Our Pieri-type formula is a generalization of Pieri’s formulas obtained by A. Lascoux and M.-P. Schützenberger for flag varieties of type A. We also introduce a super-version of the bracket algebra together with a family of pairwise anticommutative elements, the so-called flat connections with constant coefficients, which describes “a noncommutative differential geometry on a finite Coxeter group” in the sense of S. Majid.     

Almost automorphic solutions for abstract functional differential equations

For abstract linear functional differential equations with an almost automorphic forcing term, we establish a result on the existence of almost automorphic solutions, which extends the classical theorem due to Massera on the existence of periodic solutions for linear periodic ordinary differential equations.     

Ethylene polymerization by chromium complexes with phosphorous‐bridged bisphenoxy ligand

Chromium catalysts combined with phosphorous‐bridged bisphenoxy ligands were found to be highly active for ethylene polymerization. The most efficient catalyst precursor among them, generated by combining bis[3‐tert‐butyl‐5‐methyl‐2‐hydroxyphenyl](phenyl)phosphine hydrochloride ( 1a ) and CrCl₃(THF)₃, was characterized. X‐ray analysis of (3‐tert‐butyl‐5‐methyl‐2‐phenoxy)(3‐tert‐butyl‐5‐methyl‐ 2‐hydroxyphenyl)(phenyl)phosphine bis(tetrahydrofuran)chromium dichloride ( 6 ), obtained by the reaction of 1a and CrCl₃(THF)₃ in the presence of NaH, revealed a unique structure in which one phenol moiety of the bisphenol did not coordinate to the chromium center. Complex 6 showed higher activities than those observed in the in situ catalyst system. Polyethylene of various molecular weights was obtained with differing activators. The highest activity (113.5 kg mmol (cat)^?1 h^?1) was observed when TIBA/TB was used as a cocatalyst. A medium molecular weight polymer with narrow molecular weight distribution (M_w = 128,700, M_w/M_n = 1.8) was obtained using a 6 ‐TIBA/B(C₆F₅)₃ system. © 2007 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 45: 3668–3676, 2007     

Constructive and Destructive Two-Pulse Excitation Investigated with a White-Light Michelson Interferometer

Linear propagation of two pulses through methanol solution of aluminum phthalocyanine chloride is investigated using a modified white-light Michelson interferometer. The observed coherence time of the white light is 6 fs, and the separation between the two-excitation pulses is set to about 10 fs. The excitation is dependent on the phase-relation between the two pulses. We have observed an enhancement of the excitation when the two pulses are in-phase and strong suppression of the excitation when the two pulses are out of phase by π.     

Investigation of the drying conditions for amidosulfuric acid.

The drying conditions for primary standards of volumetric analysis have a significant effect on the titration results due to changes in the purity, stability and homogeneity. Amidosulfuric acid, a strong acid used as a reference material for volumetric analysis in Japan, was dried in a vacuum desiccator or heated at different temperatures, and then measured by Karl-Fischer titration, thermogravimetry/mass spectroscopy (TG-MS), ion chromatography and coulometric titration. The optimum drying conditions were at 50 degrees C for 24 h with crushing.     

Hydration Structure Around the Carboxyl Group Studied by Neutron Diffraction with 12C/13C and H/D Isotopic Substitution Methods

Time-of-Flight (TOF) neutron diffraction measurements have been carried out on aqueous 8 mol% sodium acetate solutions in D₂O. Scattering cross sections that were observed for sample solutions involving ¹²C/¹³C and H/D isotopically substituted acetate ions were used to derive the first-order difference functions, Δ_H(Q) and Δ_C(Q), and corresponding distribution functions, G _H(r;r) and G _C(r;r), which describe the environmental structure around the methyl and the carboxyl groups within the acetate ion, respectively. Structural parameters concerning the first hydration shell of the carboxyl group within the acetate ion were obtained through the least squares fit to the observed intermolecular difference function, Δ_C ^inter(Q). The nearest neighbor C _O...D _W1 (C_O: carboxyl carbon atom, D_W1: water deuterium atom) distance, r(C _O...D _W1 ), and the angle, ∠ C _O ...D _W1 -O _W (O _W : water oxygen atom), were determined to be 2.63(1) Å and 120(1)°, respectively. The coordination number, n(C _O ...D _W1 ), was obtained to be 4.0(1). These results are consistent with the hydration structure in which water molecules in the first hydration shell of the carboxyl group are hydrogen-bonded with oxygen atoms of the carboxyl group.     

In Situ Immobilization of Ultrafine Particles Synthesized in a Water/Oil Microemulsion

We investigated the in situ immobilization of ultrafine particles synthesized in a water/oil (w/o) microemulsion to silica for its possible application to supported metal catalysts. ZnS particles immobilized to silica by the ME method were consistent with those synthesized in a w/o microemulsion. Therefore, ZnS particles in a w/o microemulsion could be immobilized to silica without aggregation by this method. The relationship between the method of synthesizing Rh ultrafine particles in a w/o microemulsion and the diameter and diameter distribution of Rh particles immobilized to silica was studied. Rh-SiO(2) catalysts with a sharp diameter distribution could be prepared by immobilizing Rh-hydrazine complex particles because these complex particles would be very stable in a w/o microemulsion. The Rh particle diameters of Rh-SiO(2) catalysts prepared by changing the amount of silica produced were almost identical. Accordingly, the Rh particle diameter of Rh-SiO(2) catalysts could be controlled independently of Rh content by the ME method. Copyright 2001 Academic Press.     

Magnetic interactions in a series of paramagnetic Ln(III) complexes with a chelated imino nitroxide radical

The magnetic interactions in a new series of isostructural imino nitroxide radical lanthanide(III) complexes, [Ln(hfac)₃(IM2py)] (Ln = Gd–Yb: IM2py = 2-(2′-pyridyl)-4,5-dihydro-4,4,5,5-tetramethyl-1H-imidazoline-1-oxy; hfac = 1,1,1,5,5,5-hexafluoro-2,4-pentanedione), are examined by considering the intrinsic paramagnetic contribution of the Ln(III) ion from the corresponding [Ln(hfac)₃(pybzim)] with a diamagnetic pybzim(2-(2-pyridyl)benzimidazole) ligand; the Ln(III)–IM2py interaction being antiferromagnetic for the 4f⁷ to 4f¹³ Ln(III) complexes and negligibly small for the other complexes. This series is the first example reverse to the previous cases for the series of Ln–Cu or Ln–aminoxyl(NIT) radical (4,5-dihydro-4,4,5,5-tetramethyl-1H-imidazoline-3-oxide-1-oxy) complexes, other than only a few examples of semiquinone Ln complexes. This reverse nature of the magnetic interaction, as compared with the NIT complexes, validates the empirical approach by O. Kahn et al. [Inorg. Chem. 38 (1999) 3692; J. Am. Chem. Soc. 122 (2000) 3413] in the spin-coupled systems for a series of Ln(III) complexes.